Optimal. Leaf size=349 \[ \frac{\sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+a-c} \tanh ^{-1}\left (\frac{-\sqrt{a^2-2 a c+b^2+c^2}+a+b \cot (d+e x)-c}{\sqrt{2} \sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+a-c} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt{2} e \sqrt{a^2-2 a c+b^2+c^2}}-\frac{\sqrt{\sqrt{a^2-2 a c+b^2+c^2}+a-c} \tanh ^{-1}\left (\frac{\sqrt{a^2-2 a c+b^2+c^2}+a+b \cot (d+e x)-c}{\sqrt{2} \sqrt{\sqrt{a^2-2 a c+b^2+c^2}+a-c} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt{2} e \sqrt{a^2-2 a c+b^2+c^2}}+\frac{\tanh ^{-1}\left (\frac{2 a+b \cot (d+e x)}{2 \sqrt{a} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt{a} e} \]
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Rubi [A] time = 0.687858, antiderivative size = 349, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226, Rules used = {3701, 6725, 724, 206, 1036, 1030, 208} \[ \frac{\sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+a-c} \tanh ^{-1}\left (\frac{-\sqrt{a^2-2 a c+b^2+c^2}+a+b \cot (d+e x)-c}{\sqrt{2} \sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+a-c} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt{2} e \sqrt{a^2-2 a c+b^2+c^2}}-\frac{\sqrt{\sqrt{a^2-2 a c+b^2+c^2}+a-c} \tanh ^{-1}\left (\frac{\sqrt{a^2-2 a c+b^2+c^2}+a+b \cot (d+e x)-c}{\sqrt{2} \sqrt{\sqrt{a^2-2 a c+b^2+c^2}+a-c} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt{2} e \sqrt{a^2-2 a c+b^2+c^2}}+\frac{\tanh ^{-1}\left (\frac{2 a+b \cot (d+e x)}{2 \sqrt{a} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt{a} e} \]
Antiderivative was successfully verified.
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Rule 3701
Rule 6725
Rule 724
Rule 206
Rule 1036
Rule 1030
Rule 208
Rubi steps
\begin{align*} \int \frac{\tan (d+e x)}{\sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{x \left (1+x^2\right ) \sqrt{a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{e}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{1}{x \sqrt{a+b x+c x^2}}-\frac{x}{\left (1+x^2\right ) \sqrt{a+b x+c x^2}}\right ) \, dx,x,\cot (d+e x)\right )}{e}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{e}+\frac{\operatorname{Subst}\left (\int \frac{x}{\left (1+x^2\right ) \sqrt{a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{e}\\ &=\frac{2 \operatorname{Subst}\left (\int \frac{1}{4 a-x^2} \, dx,x,\frac{2 a+b \cot (d+e x)}{\sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{e}-\frac{\operatorname{Subst}\left (\int \frac{-b+\left (a-c-\sqrt{a^2+b^2-2 a c+c^2}\right ) x}{\left (1+x^2\right ) \sqrt{a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{2 \sqrt{a^2+b^2-2 a c+c^2} e}+\frac{\operatorname{Subst}\left (\int \frac{-b+\left (a-c+\sqrt{a^2+b^2-2 a c+c^2}\right ) x}{\left (1+x^2\right ) \sqrt{a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{2 \sqrt{a^2+b^2-2 a c+c^2} e}\\ &=\frac{\tanh ^{-1}\left (\frac{2 a+b \cot (d+e x)}{2 \sqrt{a} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt{a} e}-\frac{\left (b \left (a-c-\sqrt{a^2+b^2-2 a c+c^2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-2 b \left (a-c-\sqrt{a^2+b^2-2 a c+c^2}\right )+b x^2} \, dx,x,\frac{a-c-\sqrt{a^2+b^2-2 a c+c^2}+b \cot (d+e x)}{\sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt{a^2+b^2-2 a c+c^2} e}+\frac{\left (b \left (a-c+\sqrt{a^2+b^2-2 a c+c^2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-2 b \left (a-c+\sqrt{a^2+b^2-2 a c+c^2}\right )+b x^2} \, dx,x,\frac{a-c+\sqrt{a^2+b^2-2 a c+c^2}+b \cot (d+e x)}{\sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt{a^2+b^2-2 a c+c^2} e}\\ &=\frac{\tanh ^{-1}\left (\frac{2 a+b \cot (d+e x)}{2 \sqrt{a} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt{a} e}+\frac{\sqrt{a-c-\sqrt{a^2+b^2-2 a c+c^2}} \tanh ^{-1}\left (\frac{a-c-\sqrt{a^2+b^2-2 a c+c^2}+b \cot (d+e x)}{\sqrt{2} \sqrt{a-c-\sqrt{a^2+b^2-2 a c+c^2}} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt{2} \sqrt{a^2+b^2-2 a c+c^2} e}-\frac{\sqrt{a-c+\sqrt{a^2+b^2-2 a c+c^2}} \tanh ^{-1}\left (\frac{a-c+\sqrt{a^2+b^2-2 a c+c^2}+b \cot (d+e x)}{\sqrt{2} \sqrt{a-c+\sqrt{a^2+b^2-2 a c+c^2}} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt{2} \sqrt{a^2+b^2-2 a c+c^2} e}\\ \end{align*}
Mathematica [C] time = 35.143, size = 64621, normalized size = 185.16 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 2.547, size = 49673, normalized size = 142.3 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tan \left (e x + d\right )}{\sqrt{c \cot \left (e x + d\right )^{2} + b \cot \left (e x + d\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tan{\left (d + e x \right )}}{\sqrt{a + b \cot{\left (d + e x \right )} + c \cot ^{2}{\left (d + e x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tan \left (e x + d\right )}{\sqrt{c \cot \left (e x + d\right )^{2} + b \cot \left (e x + d\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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